Need yard help??? (Very small area)

The place we're at has a tiny fenced in area in the back. The problem we're having is that the "grassy" area is nothing but mud pretty much year round. We have dogs and the area doesn't get direct sunlight on it because of the Canopy/Awning type thing on the patio. Of course the landlord wants grass there, but it's pretty much impossible. We've tried growing it and it's only small spots where it grows. We get so much rain at times that it gets flooded so the grass always dies. As well as the dogs peeing on it doesn't help. The dogs ALWAYS track mud and dirt in the house and I can't take it anymore.

Is there anything we can do to the area to make it not muddy? I've thought about getting artificial grass put in, but since we're renting I don't want to spend that much money. We've been there 2 years now, going on a 3rd year. We had it all nice with mulch around the sides, etc. but now it's horrible. It's so bad that I wish I can put mulch over the whole area!

Re: Need yard help??? (Very small area)

What about nice gravel?